A general mass‐conservative numerical solution for the unsaturated flow equation

TLDR

Numerical approximations of unsaturated flow using different PDE forms can yield significantly different results, and the standard h‑based Richards equation often produces large mass balance errors and inaccurate infiltration depths. Accurate solutions of the unsaturated flow equation require a diagonal time (or mass) matrix. Mixed‑form Richards equation solutions are mass‑conservative and improve performance without extra cost, yet only when diagonal time matrices are used do they satisfy a maximum principle, guaranteeing smooth, non‑oscillatory infiltration profiles.

Abstract

Numerical approximations based on different forms of the governing partial differential equation can lead to significantly different results for unsaturated flow problems. Numerical solution based on the standard h ‐based form of Richards equation generally yields poor results, characterized by large mass balance errors and erroneous estimates of infiltration depth. Conversely, numerical solutions based on the mixed form of Richards equation can be shown to possess the conservative property, so that mass is perfectly conserved. This leads to significant improvement in numerical solution performance, while requiring no additional computational effort. However, use of the mass‐conservative method does not guarantee good solutions. Accurate solution of the unsaturated flow equation also requires use of a diagonal time (or mass) matrix. Only when diagonal time matrices are used can the solution be shown to obey a maximum principle, which guarantees smooth, nonoscillatory infiltration profiles. This highlights the fact that proper treatment of the time derivative is critical in the numerical solution of unsaturated flow.